| L(s) = 1 | + 2·2-s − 8.51·3-s + 4·4-s + 16.7·5-s − 17.0·6-s + 25.1·7-s + 8·8-s + 45.5·9-s + 33.5·10-s − 16.1·11-s − 34.0·12-s + 62.6·13-s + 50.3·14-s − 143.·15-s + 16·16-s + 16.4·17-s + 91.0·18-s + 58.1·19-s + 67.1·20-s − 214.·21-s − 32.3·22-s − 170.·23-s − 68.1·24-s + 156.·25-s + 125.·26-s − 157.·27-s + 100.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.63·3-s + 0.5·4-s + 1.50·5-s − 1.15·6-s + 1.35·7-s + 0.353·8-s + 1.68·9-s + 1.06·10-s − 0.443·11-s − 0.819·12-s + 1.33·13-s + 0.961·14-s − 2.46·15-s + 0.250·16-s + 0.234·17-s + 1.19·18-s + 0.702·19-s + 0.750·20-s − 2.22·21-s − 0.313·22-s − 1.54·23-s − 0.579·24-s + 1.25·25-s + 0.944·26-s − 1.12·27-s + 0.679·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.975533273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.975533273\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 269 | \( 1 - 269T \) |
| good | 3 | \( 1 + 8.51T + 27T^{2} \) |
| 5 | \( 1 - 16.7T + 125T^{2} \) |
| 7 | \( 1 - 25.1T + 343T^{2} \) |
| 11 | \( 1 + 16.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 16.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 58.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 170.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 7.21T + 2.43e4T^{2} \) |
| 31 | \( 1 + 126.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 15.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 172.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 608.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 545.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 487.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 576.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 930.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 287.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 331.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 231.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 671.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 927.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 128.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.57332085386726448489013262844, −10.06599245586433303551269540845, −8.622723638975118981391460182528, −7.36067166543168933792350378491, −6.26380355694184789909730645779, −5.55160673363993102881892139392, −5.27656549813981684901962382098, −4.03743838081287627355022616560, −2.04362751581036137811721364359, −1.15123362350823851390247428078,
1.15123362350823851390247428078, 2.04362751581036137811721364359, 4.03743838081287627355022616560, 5.27656549813981684901962382098, 5.55160673363993102881892139392, 6.26380355694184789909730645779, 7.36067166543168933792350378491, 8.622723638975118981391460182528, 10.06599245586433303551269540845, 10.57332085386726448489013262844
